Firstly, we show the difference between a real network and its corresponding null models in Figure REF a. For a network \(G\) , we consider three kinds of null models (\(k\) -order null models, including \(k\) =1.0, 2.0 and 2.5) [1]}, which is defined as \(Dk1.0\) , \(Dk2.0\)  and \(Dk2.5\) , respectively. Specifically, different values of \(k\)  indicate the preservation of network topology to different degrees. \(k=1.0\)  indicates that the generated network retains the degree sequence. When \(k=2.0\) , the degree sequence and degree correlation properties are invariant during the rewiring process. \(k=2.5\)  preserves the clustering spectrum property of the original network. The dissimilarity values are averaged over 100 repeated independent runs. With the increase of \(k\) , the dissimilarity between a real network and its randomized networks tends to be smaller across different networks (each row in Figure REF a). The pattern of the network dissimilarity values is consistent with the randomization process, where larger \(k\)  indicates that the randomized networks share more properties with the original network, leading to the more similarity to the original network.
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